Doing the first mission allows for easier supply gathering, and is necessary to lay a trap (also, not losing your queen). But does the second mission have any effect on the game? As far as I can tell, it only add an epilogue and unlock an achievement. Is there anything else?

The second mission (unless you take the route of betraying the Red Mask, which has more gameplay consequences) is more of a bonus option roleplaying choice shaping Elsa's future character. It doesn't change the regular gameplay.